Each trial can result in just two possible outcomes. We call one of these
outcomes a success and the other, a failure.

The probability of success, denoted by p, is the same on every
trial.

The trials are independent;
that is, the outcome on one trial does not affect the outcome on other trials.

The experiment continues until r successes are observed, where r
is specified in advance.

Consider the following statistical experiment. You flip a coin repeatedly and count
the number of times the coin lands on heads. You continue flipping the coin until
it has landed 5 times on heads. This is a negative binomial experiment
because:

The experiment consists of repeated trials. We flip a coin repeatedly until it
has landed 5 times on heads.

Each trial can result in just two possible outcomes - heads or tails.

The probability of success is constant - 0.5 on every trial.

The trials are independent; that is, getting heads on one trial does not affect
whether we get heads on other trials.

The experiment continues until a fixed number of successes have occurred;
in this case, 5 heads.

Suppose we flip a coin repeatedly and count the number of heads (successes).
If we continue flipping the coin until it has landed 2 times on heads, we
are conducting a negative binomial experiment. The negative
binomial random variable is the number of coin flips required to achieve
2 heads. In this example, the number of coin flips is a random variable
that can take on any integer value between 2 and
plus infinity. The negative binomial probability distribution for
this example is presented below.

Suppose that we conduct the following negative binomial
experiment. We
flip a coin and count the number of flips until the coin has landed
three times on Heads. If we need to flip the coin 5 times until the coin
has landed on Heads 3 times, then 5
is the number of trials.

What is the number of successes?

Each trial in a negative binomial experiment can have one of two outcomes.
The experimenter classifies one outcome as a success; and the other, as a
failure. The number of successes in a binomial experient is the number of
trials that result in an outcome classified as a success.

What is the probability of success on a
single trial?

In a negative binomial experiment, the probability of success on any
individual trial is constant. For example, the probability of getting Heads on
a single coin flip is always 0.50. If "getting Heads" is defined as success,
the probability of success on a single trial would be 0.50.

For example, suppose we conduct a
negative binomial experiment to count the number of coin flips
required for a coin to land 2 times on Heads. We might ask: What is
the probability that this experiment will require 5 coin flips?
In this example, we would be asking about a negative binomial probability.
(From the above table, you can see that the probability that this
experiment would require 5 coin flips is 0.125.)

What is the relation between a
binomial experiment and a negative binomial experiment?

With a binomial experiment, we are concerned with finding
the probability of r successes in x trials, where x
is fixed. With a negative binomial experiment, we are concerned with
finding the probability that the rth success occurs on the
xth trial, where r is fixed.

What is the relation between a
geometric distribution and a negative binomial distribution?

The geometric distribution is a special case of the
negative binomial distribution. It deals with the number of trials
required for a single success. Thus, the geometric distribution is
negative binomial distribution where the number of successes (r)
is equal to 1.

With a negative binomial distribution, we are concerned with
finding the probability that the rth success occurs on the
xth trial, where r is fixed. With a
geometric distribution, we are concerned with
finding the probability that the first success occurs on the
xth trial.

Can I use the
Negative Binomial Calculator
to solve problems based on the geometric distribution?

Of course! The geometric distribution is just a special
case of the negative binomial distribution (see above question);
so geometric distribution problems can be solved with the
Negative Binomial Calculator.

Negative Binomial Distribution: Sample Problems

The probability that a driver passes the written test for a driver's
license is 0.75. What is the probability that a person will fail the
test on the first try and pass the test on the second try?

Solution:

We know the following:

The number of trials is 2.

The number of successes is 1 (since we define passing the test as success).

The probability of success (i.e., passing the test) on any single trial is 0.75.

Therefore, we plug those numbers into the
Negative Binomial Calculator
and hit the Calculate button. The calculator reports that the negative binomial
probability is 0.1875. That is the probability of failing the first test
and passing the second test.

Find the probability that a man flipping a coin gets the fourth head on the
ninth flip.

Solution:

We know the following:

The number of trials is 9 (because we flip the coin nine times).

The number of successes is 4 (since we define Heads as a success).

The probability of success for any coin flip is 0.5.

Therefore, we plug those numbers into the
Negative Binomial Calculator
and hit the Calculate button. The calculator reports that
the negative binomial probability is 0.109375. That is the probability that
the coin will land on heads for the fourth time on the ninth coin flip.